Propositional logic is a good vehicle to introduce basic properties of logic

Logic in computer sience
Propositional logic is a good vehicle to introduce basic properties of logic. It does not provide
means to determine the validity (truth or false) of atomic statements. Instead, it allows you to
evaluate the validity of compound statements given the validity of its atomic components.
For example, consider the following:
I like Pat or I like Joe.
If I like Pat then I like Joe.
Do I like Joe?
Accept as facts the first two statements, noting that the use of "or" here is not exclusive and thus
could really be thought of as saying "I like Pat, or I like Joe, or I like them both". Do these
statements imply that "I like Joe" is true? Try to convince yourself that "I like Joe" is true, and
consider another line of reasoning:
Pigs can fly or fish can sing.
If pigs can fly then fish can sing.
Can fish sing?
We can see that the answer is yes in both cases. The above two sets of statements can be both
abstracted as follows:
Q?
Here, we are concerned about the logical reasoning itself, and not the statements. Thus, instead
of working with pigs or Pats, we simply write Qs or Ps. We begin our study first with the syntax
of propositional logic: that is, we describe the elements in our language of logic and how they are
written. We then describe the semantics of these symbols: that is, what the symbols mean.
[edit] Syntax
The syntax of propositional logic is composed of propositional symbols, logical connectives, and
parenthesis. Rules govern how these elements can be written together. First, we treat
propositional symbols merely as a set of some symbols, for our purposes we'll use letters of the
Roman and Greek alphabets, and refer to the set of all symbols as Prop:
Propositional symbols: A set Prop of some symbols. For example
Second, we have the logical connectives:

Logical connectives:
Note that these are not the minimal required set; they can be equivalently represented only using
the single connective NOR (not-or) or NAND (not-and) as is used at the lowest level in computer
hardware. Finally, we use parenthesis to denote expressions (later on we make parenthesis
optional):
Parentheses: (,)
An expression is a string of propositional symbols, parenthesis, and logical connectives.
The expressions we consider are called formulas. The set Form of formulas is the smallest set of
expressions such that:
1.
2. If then
1. ,
2. ,
3. , and
4. .
Another way to define formulas is as the language defined by the following context-free
grammar (with start symbol Form):
, where Prop stands for any propositional symbol
Fact 1 (Unique Readability): The above context free grammar is unambiguous.
[edit] Semantics
The function of a formula is to create meanings of statements given meanings of atomic
statements. The semantics of a formula φ with propositional symbols is a mapping
associating to each truth assignment V to a truth value (0 or 1) for φ. (The truth
values true and false can be used instead of 1 or 0, respectively, as well as the abbreviations T
and F.)
The semantics are well defined due to Fact 1.
One way to specify semantics of a logical connective is via a truth table:

p q
0 0 0
0 1 0
1 0 0
1 1 1
Can one always find a formula that implements any given semantics? Yes, any truth table is
realized by a formula. The formula can be found as follows. "Represent" the rows where φ = 1
with conjunctions of the true proposition symbols and negations of the false ones. Finally write
the disjunction of the results.
For example,
p q φ Conjunctions (true values only)
0 0 1
0 1 0
1 0 1
1 1 0
Corollary: Every formula is equivalent to a disjunction of conjunctions of propositional symbols
or negation of propositional symbols (DNF).
Dual of DNF is CNF. To get φ in CNF:
1. Describe cases when φ is false. ex - DNF ψ
2. Note that φ is true when is false. Hence, negate ψ using DeMorgan's laws.
ex .
There are cases when DNF (resp. CNF) is exponentially larger than the original formula. For
example, for the equivalent DNF is exponential
in size.
Does each truth table have a polynomial size formula implementing it? More precisely, does
there exist k such that every truth table with n propositional symbols has a form φ of size ?
Answer: no.

Proof: Assume there exists such k. The number of truth tables for n propositional symbols is
. The number of formulas of size is (n propositional symbols, 4
connectives and parentheses.) Clearly, , for sufficiently large n.
[TODO: exposition to explain what these definitions are and provide their context]
 Satisfaction: Satisfaction of a formula φ by a truth assignment τ. Notation: (φ is
true for τ).
 Implication: A set of formulas Σ implies φ. Notation: . Σ implies φ if and only if
every truth assignment that satisfies Σ also satisfies φ.
[edit] Formula Classes of Special Interest
 VALID - the set of formulas that are always true (also known as tautologies). For
example, are valid formulas.
 UNSAT - the set of formulas that are never true (unsatisfiable).
 In between: SAT - the set of formulas for which there exists a satisfying assignment (not
unsatisfiable).
Note. .
Claim:
Claim: SAT is NP-complete.
Proof:
 : guess a satisfying assignment, then verify that the formula is true (a
satisfying assignment is a certificate).
 Hardness. graph 3-coloring (there also exists a direct proof). We reduce 3-
coloring to SAT. Let G = (V,E) be a graph with n nodes . We use
propositional variables pi,g,pi,r,pi,b to indicate that vertex i is colored with green, red, or
blue. Construct φ as follows:
Claim: .
It is also possible to prove that directly

Claim: .
[edit] Horn Clauses
Special case for which SAT is in polynomial time. Example:
A Horn clause is a disjunction of literals of which at most one is positive. There are two kinds of
possible Horn clauses:
1. clause has 1 positive literal
1. p, or
2.
2. no positive literal
1.
2.
Claim: For every set Σ of Horn formulas, checking whether Σ is satisfiable is in P.
Proof Idea: Let Σ1 be the subset of Σ containing only clauses of type 1, and Σ2 the subset of Σ
containing clauses of type 2. Note first that Σ1 is satisfiable. To obtain a minimum satisfying
assignment σ, start with literals from single-literal clauses and crank the rules. It now remains to
check consistency of σ with the clauses in Σ2. To do this, it is enough to check that for each
clause in Σ2, σ is not true for all of .
Example: Consider the set Σ of Horn clauses:
p
q
r
The set Σ1 of clauses of type 1 consists of the first 5 clauses, and Σ2 consists of the last clause.
Note that Σ1 can also be written as:
p
q
r
The minimum satisfying assignment for Σ1 is obtained as follows:

1. start with {p,q,r}
2. use the first implication to infer s
3. use the second implication to infer t
Thus, the minimum satisfying assignment makes {p,q,r,s,t} true. This contradicts Σ2, which
states that t must be false. Thus, Σ is not satisfiable.
[edit] Deductive Systems
A deductive system is a mechanism for proving new statements from given statements.
Let Σ be a set of known valid statements (propositional formulas). In a deductive system, there
are two components: inference rules and proofs.
Inference rules
An inference rule indicates that if certain set of statements (formulas) is
true, then a given statement must be true. An inference rule H is denoted as
.
Example (modus ponens):
Proofs
A proof of from Σ is sequence of formulas such that and for all
 Each formula , or
 There are a subset of formulas , such that,
is an inference rule.
If has a proof from Σ using inference rule H we write .
Properties:
 Soundness: If then (i.e., all provable sentences are true). This property
is fundamental for the correctness of the deductive system.
 Completeness: If then (i.e., all true sentences are provable). This is a
desirable property in deductive systems.
[edit] Natural Deduction
Natural deduction is a collection of inference rules. Let denote contradiction, falsity. The
following are the inference rules of natural deduction:

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

14.
15.
16.
Rule (13) allows us to prove valid statements of the form "If then ψ" even if we don't know the
truth value of the statement (i.e., is not in the set Σ of known valid statements). Indeed, for
this rule, we start assuming is valid. If we can conclude ψ is valid in a world where are
valid, then we conclude that the relation is true, and we "release" the assumption is
valid.
We now show how to apply the above inference rules.
Example: De Morgan's Law for negated or-expressions says:
Proof: By rule (8) if we can prove and
we can infer the desired result.
To prove the first direction, we use rule 13 and assume the hypothesis . Then

(assumed)
(assumed)
(by rule 11)
(by rule 5)
(by rule 14)
ψ (assumed)
(by rule 11)
(by rule 5)
(by rule 14)
(by rule 1)
(by rule 13)
We now prove the second direction.
(assumed)
(by rule 2)
(by rule 3)
(assumed)
(assumed)
(by rule 5)
(by rule 16)
(by rule 14)
(by rule 13)
Proof of Pierce's Law:
.
(assumed) (1*)
(assumed)
A (assumed)
(by rule 5)
B (by rule 7)
(by rule 13)
A (by assumption (1*) and rule 4)
(by rule 5)
A (by rule 14)
(by rule 13)
Fact 2: Natural deduction is sound.
To show that natural deduction is also complete we need to introduce propositional resolution.
[edit] Propositional Resolution

Resolution is another procedure for checking validity of statements. It involves clauses,
formulas and a single resolution rule.
Some terminology:
Clause
A clause is a propositional formula composed by disjunction of literals. For example
. It is usually denoted as the set of literals, e.g. .
The empty clause, denoted as an open box " ", is the disjunction of no literals. It is
always false.
Formula
A set of clauses, each of them satisfiable. For example,
represents the CNF formula .
The empty formula, denoted as , is the set that contains no clauses. It is always true.
Resolution Rule
It is a rule that, given two clauses C (containing some literal y) and C' (containing some
literal ), allows to infer a new clause, called the resolvent of C and C' (with respect to
y).
A proof system for resolution contains a single resolution rule, where the resolvent is defined as
follows. Assume C and C' are clauses such that and , then
.
The smallest set of clauses containing and closed under resolution is denoted .
Example: If C = {p,y} and , then resy(C,C') = {p,q}.
It is possible to show that the resolution rule, as defined, computes a clause that can be inferred
using natural deduction.
Claim: Let C and C' be any two clauses such that and . Then
.
In order to prove the validity of a statement ψ, we will prove the negated statement is
unsatisfiable. To prove unsatisfiability of a formula , we need to define the resolution
refutation of the formula :
The resolution refutation tree of the formula is a tree rooted at the empty clause, where every
leaf is a clause in and each internal node is computed as the resolvent of the two
corresponding children.
Notice that clauses of can appear repeated as leaves. From above claim we can conclude that:

Claim: If there exists a resolution refutation tree for formula , then , that is, is
unsatisfiable.
Example: The formula
has the following resolution refutation tree:
The order in which clauses are selected to compute the resolvent matters when computing the
resolution refutation tree, as the following example shows: Consider the formula
.
Even though a resolution refutation tree may exist for ψ, order is important when trying to build
the tree. Below are two different resolution refutation trees, but only one is successful:
Unsuccessful attempt of resolution refutation tree for ψ.

A successful resolution refutation tree for ψ.
[edit] Properties of Propositional Resolution
Soundness: Propositional resolution is sound, that is, if there exists a resolution refutation tree
for a given formula , then must be unsatisfiable.
Theorem: For any formula , if , then .
Completeness: Propositional resolution is complete, that is, if a given formula is unsatisfiable,
then has a resolution refutation tree.
Theorem: For any formula , if , then .
Proof: By induction on the number of variables in .
Basis: We have one variable, say p. All possible clauses of are {p} and . If is
unsatisfiable then both clauses occur, and therefore .
Induction step: Suppose the hypothesis is true for formulas with less than n variables. Let be a
formula with n variables. Suppose ; we will show is satisfiable. Let p be a
variable of . Then either or (if both hold then
immediately).
Assume . We define the formula as containing all clauses that do not
contain {p} and where the literal has been removed from each clause (in other words, is
equivalent to the formula resulting from setting p true).
Formally,
.
First, notice that

and thus,
.
Also, since we have that . By the induction hypothesis, is
satisfiable. Then is satisfiable by an extension of the satisfying assignment of with p equal
true. The case is analogous.
[edit] Completeness of Natural Deduction
Theorem: Let H be the set of inference rules of Natural Deduction. If then .
The idea behind the proof of completeness of natural deduction is as follows. Suppose is valid
(then is unsatisfiable). We then show there exists a resolution refutation for and then by
applying the contradiction rule (rule 15):
we conclude can be inferred.
Proof: (Sketch) Given a formula valid under Σ, we perform the following steps:
1. Prove that is equivalent to some ψ, where ψ is in CNF.
2. Prove that , for all ψ.
3. By completeness of resolution, if ψ is unsatisfiable then . Therefore, {p}
and for some literal p. This implies .
4. Conclude that and therefore is valid.
Step (1) can be easily done by repeated application of De Morgan's laws. Step (2) can be proven
using natural deduction. Finally, step (3) can be proven by induction on the number of steps to
obtain Res(ψ). Clearly, each step can be simulated using natural deduction.
It is very likely that any algorithm for propositional resolution will take very long on the worst
case (recall that checking validity of a formula is co-NP complete).
[edit] Linear Resolution and PROLOG

Linear resolution is a particular resolution strategy that always resolves the most recent resolvent
with a clause. The resolution refutation tree so obtained is therefore linear. It is possible to prove
that, if the set of clauses are Horn clauses, there exists a linear resolution strategy for any
formula. That is, linear resolution is complete for the set of Horn clauses.
The language PROLOG uses resolution on a set of Horn clauses. Each clause is called a
program clause. Moreover, clauses composed by a single literal are called facts. A clause with a
single negated literal is called a query. The table below shows a comparison of the different
notations. In PROLOG, to query a statement t, the idea is to negate the statement ( ) and to
perform resolution with the set of known true statements. If a resolution refutation tree is found,
the statement t is implied by the program.
Example: An example of linear resolution for the formula
is shown here:

Propositional logic is a good vehicle to introduce basic properties of logic

Propositional logic is a good vehicle to introduce basic properties of logic

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Propositional logic is a good vehicle to introduce basic properties of logic